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Theorem dftpos2 8253
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
dftpos2 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos2
StepHypRef Expression
1 dmtpos 8248 . . 3 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
21reseq2d 5987 . 2 (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹dom 𝐹))
3 reltpos 8241 . . 3 Rel tpos 𝐹
4 resdm 6033 . . 3 (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹)
53, 4ax-mp 5 . 2 (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹
6 df-tpos 8236 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
76reseq1i 5983 . . 3 (tpos 𝐹dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹)
8 resco 6257 . . 3 ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹))
9 ssun1 4172 . . . . 5 dom 𝐹 ⊆ (dom 𝐹 ∪ {∅})
10 resmpt 6044 . . . . 5 (dom 𝐹 ⊆ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥}))
119, 10ax-mp 5 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥})
1211coeq2i 5865 . . 3 (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹)) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
137, 8, 123eqtri 2759 . 2 (tpos 𝐹dom 𝐹) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
142, 5, 133eqtr3g 2790 1 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cun 3945  wss 3947  c0 4324  {csn 4630   cuni 4910  cmpt 5233  ccnv 5679  dom cdm 5680  cres 5682  ccom 5684  Rel wrel 5685  tpos ctpos 8235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-tpos 8236
This theorem is referenced by:  tposf12  8261
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